Problem: Simplify; express your answer in exponential form. Assume $a\neq 0, z\neq 0$. $\dfrac{{(a^{4}z^{-5})^{5}}}{{(a^{4}z^{-2})^{-2}}}$
Answer: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(a^{4}z^{-5})^{5} = (a^{4})^{5}(z^{-5})^{5}}$ On the left, we have ${a^{4}}$ to the exponent ${5}$ . Now ${4 \times 5 = 20}$ , so ${(a^{4})^{5} = a^{20}}$ Apply the ideas above to simplify the equation. $\dfrac{{(a^{4}z^{-5})^{5}}}{{(a^{4}z^{-2})^{-2}}} = \dfrac{{a^{20}z^{-25}}}{{a^{-8}z^{4}}}$ Break up the equation by variable and simplify. $\dfrac{{a^{20}z^{-25}}}{{a^{-8}z^{4}}} = \dfrac{{a^{20}}}{{a^{-8}}} \cdot \dfrac{{z^{-25}}}{{z^{4}}} = a^{{20} - {(-8)}} \cdot z^{{-25} - {4}} = a^{28}z^{-29}$